Computer Science 302 Spring 2017 (Practice for) Final Examination, May 10, 2017

Transcription

1 Computer Science 302 Spring 2017 (Practice for) Final Examination, May 10, 2017 Name: The entire practice examination is 1005 points. 1. True or False. [5 points each] The time to heapsort an array of n items is O(nlogn). Open hashing uses open addressing. In the decision tree model of computation, the time complexity of any algorithm to sort n items is Ω(nlogn). (d) (e) (f) The height of a binary tree with n nodes is O(logn). A binary search tree is commonly used to represent unfulfilled obligations. Quicksort takes O(nlogn) time to sort an array of n items. (g) Given the choice between two algorithms, one of which takes O(n) time and the other of which takes O(n 2 ) time, it is always best to choose the one which takes O(n) time. (h) (i) (j) (k) (l) The built-in function random is an excellent choice for a hash function. A queue is typically used to store unfulfilled obligations. A stack is typically used to store unfulfilled obligations. A heap is typically used to store unfulfilled obligations. A hash table is typically used to store unfulfilled obligations. 2. [10 points] What search structure should you use if the average number of items that will be in the structure at any given time is two? 3. [5 points] What are the smallest and largest possible number of nodes that a 2-3 tree with height 3 could have? and. 4. [5 points] You cannot use algorithm if any edge has negative weight. 5. [10 points] What implementation of the ADT search structure would you use if the expected number of items in the structure is 1? 6. [5 points] Push and pop are operators of the ADT. 1

2 7. [10 points] and are operators of the ADT array. 8. [30 points]each of the following answers is from this list: greedy, divide and conquer, dynamic programming. How would you describe Kruskal s algorithm? How would you describe mergesort? How would you describe quicksort? (d) How would you describe binary search? 9. [10 points each] For each of the following code fragments, express the asymptotic time complexity by choosing the best of the following answers: O(n), O(n 2 ), O(nlogn), O(logn), O(loglogn), Θ(n), Θ(n 2 ), Θ(nlogn), Θ(logn), Θ(loglogn) (d) (e) for (int i = 0; i < n; i = 2*i+1) for (int j = i; j > 0; j = j/2); for (int j = n; j > i/2; j = j/2); for (int i = 0; i < n; i = i*i+1) 10. [30 points] Describe the meaning of the word collision as used in discussions of hashing. How are collisions handled in closed hashing? How are collisions handled in open hashing? 11. [10 points] What implementation of the ADT search structure would you use if n items are to be inserted at once at the beginning of the program, there will be no further inserts, and find will be executed n 2 times during the running of the program? (There is more than one correct answer to this problem, as well as several inferior answers.) 12. [20 points] Walk through the steps of the stack algorithm used to evaluate the following postfix expression, showing the stack at each step: (Hint: there will be approximately 9 illustrations of the stack.) * 2 3 * - 2

3 13. [20 points] Find an optimal prefix code for the alphabet {A,B,C,D,E,F,G,H, if the frequencies of the symbols are as given in the following table: A 35 B 7 C 32 D 5 E 16 F 4 G 11 H [30 points] The Partition step of Quicksort has a loop invariant. Give that loop invariant, and illustrate its meaning by drawing a figure, or figures. 15. [30 points] Describe each of the following types of search. (Be sure to say what the structure is that is being searched in each case.) Linear search. Binary search. 16. [20 points] Write pseudocode for the array implementation of the ADT stack of integer. Your code should include procedures that implement pop, push, and empty. 17. [20 points] Explain how you would insert and delete from a stack, given that you are using singly linked nodes. Draw pictures. 18. [25 points] Explain how insertion works in a 2,3-tree. Hint: the phrase node splitting or the equivalent must be in your explanation. 19. [30 points] Given the following: class BST{ // Binary Search Tree public: BST(int); // initializes item field to parameter, links to 0 void static postorderwrite(bst*); // all items to standard ostream inorder void static insert(int, BST*&); // inserts parameter, if not there bool find(int, BST*); // parameter is in the binary search tree private: int item; // the value stored in the node BST * left; // pointer to the left subtree BST * right; // pointer to the right subtree ; 3

4 Complete the following code by writing exactly three lines: void BST::postorderWrite(BST * t){ if (t!= 0){ // Your three lines go here. 20. [30 points] Suppose you are writing a dynamic programming algorithm to find the minimum weight path between a given source vertex S and a given target vertex T in a weighted directed acyclic graph G. Describe the subproblems. In what order would you work the subproblems? 21. True or False. [5 points each] Quicksort takes O(n log n) expected time to sort an array of n items, provided randomization is used to pick the pivot items. 22. [20 points] Draw before and after figures illustrating left rotation. 23. True or False. [5 points each] The time to heapsort an array of n items is Θ(nlogn). Open hashing uses open addressing. The height of a binary tree with n nodes is Ω(logn). 24. [10 points] What implementation of the ADT search structure would you use if n items are to be inserted at once at the beginning of the program, there will be no further inserts, and find will be executed n 2 times during the running of the program? (There is more than one correct answer to this problem, as well as several inferior answers.) 25. [20 points] Explain how you would insert and delete from a stack, given that you are using the standard array implementation of stack. 26. [30 points] Describe each of the following types of search. (Be sure to say what the structure is that is being searched in each case.) 4

5 Breadth first search. Depth first search. 27. [40 points] Give four ways to implement the ADT search structure. (For example, binary search tree is one way, so don t use that one.) Just give the names of the implementations, not any details. 28. [10 points] You can sort a set of items using a binary search tree, as follows: Start with an empty binary search tree, insert the items one at a time, then visit the nodes of the tree in inorder, writing out the items. This algorithm is essentially the same as which one of these well-known sorting algorithms? (Choose one answer.) Quicksort Heapsort Mergesort 29. [10 points each] For each of the following code fragments, express the asymptotic time complexity by choosing the best of the following answers: zero, O(n), O(n 2 ), O(nlogn), O(logn), O(loglogn), Θ(n), Θ(n 2 ), Θ(nlogn), Θ(logn), Θ(loglogn), infinite loop: never ends. (d) (e) for (int i = 0; i < n; i := i+2) for (int i = 0; i < n; i = 2*i) for (int i = 0; i < 0; i++) for (int j = i; j > 0; j = j/2); for (int j = i; j > i/2; j = j/2); for (int i = 1; i < n; i = i*i) 30. [20 points] Write C++ code for the find portion of union-find. Be sure to use path compression. Do not include any other part of the program. If you write more than 10 lines, you ve written far too much. 31. [30 points] Given the following: class BST{ // Binary Search Tree public: BST(int); // initializes item field to parameter, links to 0 void static preorderwrite(bst*); // all items to standard ostream inorder void static insert(int, BST*&); // inserts parameter, if not there 5

6 bool find(int, BST*); // parameter is in the binary search tree private: int item; // the value stored in the node BST * left; // pointer to the left subtree BST * right; // pointer to the right subtree ; Complete the following code by writing exactly three lines: void BST::preorderWrite(BST * t){ if (t!= 0){ // Your three lines go here. 32. [10 points each] For each of the following code fragments, express the asymptotic time complexity by choosing the best of the following answers: O(n), O(n 2 ), O(nlogn), O(logn), O(loglogn). (d) (e) for (int i = 0; i < n; i = 2*i+1) for (int j = i; j > 0; j = j/2); for (int j = i; j > i/2; j = j/2); for (int i = 0; i < n; i = i*i+1) 33. [30 points] Describe the meaning of the word collision as used in discussions of hashing. How are collisions handled in closed hashing? How are collisions handled in open hashing? 34. [10 points] What implementation of the ADT search structure would you use if n items are to be inserted at once at the beginning of the program, there will be no further inserts, and find will be executed n 2 6

7 times during the running of the program? (There is more than one correct answer to this problem, as well as several inferior answers.) 35. [20 points] Explain how you would implement a sparse array using a search structure. Do not give any details whatsoever about the search structure itself, since that s not the point of this question. 36. [20 points] Explain how you would insert and delete from a queue, given that you are using singly linked nodes in a circular linked list implementation. Draw pictures. 37. [30 points] Given the following: class BST{ // Binary Search Tree public: BST(int); // initializes item field to parameter, links to 0 void static inorderwrite(bst*); // all items to standard ostream inorder void static insert(int, BST*&); // inserts parameter, if not there bool find(int, BST*); // parameter is in the binary search tree private: int item; // the value stored in the node BST * left; // pointer to the left subtree BST * right; // pointer to the right subtree ; Complete the following code by writing exactly three lines: void BST::inorderWrite(BST * t){ if (t!= 0){ // Your three lines go here. 38. [30 points] Use polyphase mergesort to sort the following list: A,L,G,O,R,I,T,H,M,S,R,F,U,N. Show all steps. 39. [20 points] Let H be a trep, where the alphabetically smallest item is at the root. Execute the following commands, showing the treap at each step. Insert(Bob,5) Insert(Tim,3) Insert(Ann,1) (d) Delete(Ann) 7

8 40. [30 points] Under what conditions could you use dynamic programming to solve the single source minimum path problem? Write pseudocode for dynamic programming for the single source minimum path problem. 41. [30 points] Under what conditions could you use the Bellman-Ford algorithm? Write pseudocode for the Bellman-Ford algorithm. 42. [30 points] Under what conditions could you use the Floyd-Warshall algorithm? Write pseudocode for the Floyd-Warshall algorithm. 8